ENZYME KINETICS A MODERN APPROACH – PART 5 pptx

7.13 to Vmax –[A] experimental datausing nonlinear regression, it is possible to obtain estimates of Vmax and concentra-KBA.Kalso displays a hyperbolic dependence on substrate A tion Fig

NEW METHOD OF DETERMINING pK VALUES OF CATALYTIC GROUPS 85

−1.0

−0.5 0.0 0.5 1.0

Figure 6.4 Variation in the slope of the (a) log Vmax , (b) log Vmax/K s and (c) − log K s

versus pH plots as a function of pH.

86 pH DEPENDENCE OF ENZYME-CATALYZED REACTIONS

Consider the expression for the hydrogen ion dependence of theK s of anenzyme-catalyzed reaction:

NEW METHOD OF DETERMINING pK VALUES OF CATALYTIC GROUPS 87

2 3 4 5 6 7 8 9 10

−1.0

−0.5 0.0 0.5 1.0

pKe1=5.8 pKe2=6.9

pKe1=5.7 pKe2=6.8

pH (a)

1 2 3 4 5 6

Figure 6.6 (a) pH dependence of the slope of a log Vmax/K sversus pH data set (b) pH

dependence of a logVmax/K s versus pH data set.

A logarithmic transformation of Eq (5.18), results in the expression

− log K∗

s = − logK s K es1

K e1 − log([H+]2+ K e1[H+]+ K e1 K e2 )

+ log([H+]2+ K es1[H+]+ K es1 K es2 (6.19)

The first derivative of Eq (6.19) as a function of− log[H+] (i.e., pH) is

d (− log K s∗)

d (pH) = 2[H+]2+ K e1[H+]

[H+]2+ K e1[H+]+ K e1 K e2

− 2[H+]2+ K es1[H+][H+]2+ K es1[H+]+ K es1 K es2 (6.20)

It is not as easy to calculate a value for this derivative at [H+]= K, since

the exact value will depend not only on the relative magnitude of K e1

88 pH DEPENDENCE OF ENZYME-CATALYZED REACTIONS

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5

−1.0

−0.5

0.0 0.5 1.0 1.5

Kes Ke

−logK s * logV*max/Ks*

logV*max/Ks*

logV*max

logV*max

pH (a)

Figure 6.7 (a) Simulation of the pH dependence of the logarithm of the catalytic

param-etersVmax , Vmax/K s, andK s for a monoprotic enzyme (b) Variation in the slope of the

logVmax , logVmax/K s, and− log Ks versus pH plots as a function of pH for a tic enzyme.

monopro-versusK e2, but also ofK es1 versusK es2 We do not recommend workingwith this expression, since the results obtained can be ambiguous.Caution must be exercised when using this approach to determinethe pK values of the catalytic groups since considerable error can be

introduced in their determination if they happen to be numerically close.Figure 6.5(a) is a simulation of log10(Vmax∗ /K s∗) or log10Vmax∗ versus

pH patterns as a function of the closeness between K1 and K2 values.Figure 6.5(b) shows the error between actual and predicted pK values as

a function of the difference between pK values Our simulation shows

NEW METHOD OF DETERMINING pK VALUES OF CATALYTIC GROUPS 89

that as long at the difference between pK values is greater than 1 pH

unit, the error introduced in the determination of pK values will be less

than 0.1 pH unit

Figure 6.6(a) shows an actual analysis of the pH dependence of Vmax∗ /

K s∗ for the hydration of fumarate by the enzyme fumarase The slope of

the line at the midpoint between two subsequent pH values was calculated

from the data as

In our experience, drawing straight lines through the usual small ber of data points, as carried out in the Dixon analysis, was not easy,particularly for the slope= 0 line This ambiguity made it difficult tohave confidence in the pK values determined The procedure developed

num-in this chapter is more reliable On the other hand, the pK values obtained

using the Dixon analysis and the analysis presented in this chapter werefound to be similar (Fig 6.6b).

Before leaving this topic, we would like to draw to the attention of thereader that many enzymes may have only one ionizable group among theircatalytic groups For this case, the patterns obtained for the pH dependence

of the catalytic parameters will be half that of their two-ionizable-groupcounterparts (Fig 6.7) For this case, the determination of pK e and pK es

values is less prone to error since there is no interference from a secondionizable group

CHAPTER 7

TWO-SUBSTRATE REACTIONS

Up to this point, the kinetic treatment of enzyme-catalyzed reactions hasdealt only with single-substrate reactions Many enzymes of biologicalimportance, however, catalyze reactions between two or more substrates.Using the imaginative nomenclature of Cleland, two-substrate reactions

can be classified as ping-pong or sequential In ping-pong mechanisms,

one or more products must be released before all substrates can react

In sequential mechanisms, all substrates must combine with the enzyme

before the reaction can take place Furthermore, sequential mechanisms

can be ordered or Random In ordered sequential mechanisms, substrates react with enzyme, and products are released, in a specific order In ran-

dom sequential mechanisms, on the other hand, the order of substrate

combination and product release is not obligatory These reactions can

be classified even further according to the molecularity of the kineticallyimportant steps in the reaction Thus, these steps can be uni (unimolecu-lar), bi (bimolecular), ter (termolecular), quad (quadmolecular), pent (pen-tamolecular), hexa (hexamolecular), and so on This molecularity appliesboth to substrates and products Using Cleland’s schematics, examples

of ping-pong bi bi, ordered-sequential bi bi, and random-sequential bi bireactions are, respectively,

Q

EE

(7.1)

90

Kinetic analysis of multiple substrate reactions could stop at this point.However, if more in-depth knowledge of the mechanism of a particularmultisubstrate reaction is required, a more intricate kinetic analysis has to

be carried out There are a number of common reaction pathways throughwhich two-substrate reactions can proceed, and the three major types arediscussed in turn

For the random-sequential bi bi mechanism, there is no particular order

in the sequential binding of substrates A or B to the enzyme to form theternary complex EAB A general scheme for this type of reactions is

In this model we assume that rapid equilibrium binding of either substrate

A or B to the enzyme takes place For the second stage of the reaction,equilibrium binding of A to EB and B to EA, or a steady state in theconcentration of the EAB ternary complex, may be assumed

The rate equation for the formation of product, the equilibrium tion constant for the binary enzyme–substrate complexes EA and EB (KA

dissocia-s

andK sB), the equilibrium dissociation (K s) or steady-state Michaelis (K m)constants for the formation of the ternary enzyme–substrate complexesEAB (KAB and KBA), and the enzyme mass balance are, respectively,

[ET]= [E] + [EA] + [EB] + [EAB] (7.9)

A useful relationship exists among these constants:

v

KA

s KAB+ KAB[A]+ KBA[B]+ [A][B] (7.11)where Vmax= kcat[ET]

RANDOM-SEQUENTIAL Bi Bi MECHANISM 93

For the case where the concentration of substrate A is held constant,

Eq (7.11) can be expressed as

s , KAB, and

KBA Vmax
displays a hyperbolic dependence on substrate A tion (Fig 7.1a) Thus, by fitting Eq (7.13) to Vmax
–[A] experimental datausing nonlinear regression, it is possible to obtain estimates of Vmax and

concentra-KBA.K
also displays a hyperbolic dependence on substrate A tion (Fig 7.1b) However, this hyperbola does not go through the origin.

of KAB, KBA, and K sB (y-intercept) are known, it is straightforward to

obtain an estimate ofK sA using Eq (7.10):

K sA = K sBKBA

KAB (7.15)

For the case where the concentration of substrate A is held constant,

Eq (7.11) can be expressed as

Figure 7.1 Fixed substrate concentration dependence for enzymes displaying

random-sequential mechanisms: (a) Dependence of Vmax
on [A]; (b) dependence of K
on [A]; (c) dependence of Vmax
on [B]; (d) dependence of K
on [B].

From determinations of K
and Vmax
at different fixed concentrations of

substrate B, it is possible to obtain estimates of Vmax, K sB, KAB, and

KBA Vmax
displays a hyperbolic dependence on substrate B tion (Fig 7.1c) Thus, by fitting Eq (7.17) to Vmax
–[B] experimental datausing nonlinear regression, it is possible to obtain estimates of Vmax and

concentra-KAB.K
also displays a hyperbolic dependence on substrate B tion (Fig 7.1d) However, this hyperbola does not go through the origin.

For this mechanism, the enzyme must bind substrate A first, followed

by binding of substrate B, to form the ternary complex EAB A generalscheme for this type of reactions is

dissoci-v = kcat[EAB] (7.21)

K sA = [E][A]

[EA] KAB = [EA][B]

[ET]= [E] + [EA] + [EAB] (7.23)

Normalization of the rate equation by total enzyme concentration (v/[E T])

and rearrangement results in the rate equation for ordered-sequential bi

For the case where the concentration of substrate B is held constant,

Eq (7.24) can be expressed as

v

Vmax

= [A]

From determinations of K
and Vmax
at different fixed concentrations of

substrate B, it is possible to obtain estimates ofVmax,K sA, andKAB.Vmax
displays a hyperbolic dependence on substrate B concentration (Fig 7.2a).

Thus, by fitting Eq (7.26) toVmax
–[B] experimental data using nonlinearregression, it is possible to obtain estimates of Vmax and KAB K
alsodisplays a hyperbolic dependence on substrate B concentration (Fig 7.2b).

However, the y-intercept ([B] = 0) of this hyperbola equals K sA, while

in the limit where [B] approaches infinity, K
= 0 (Fig 7.2b) Thus, by

fitting Eq (7.27) toK
–[B] experimental data using nonlinear regression,

it is possible to obtain an estimate ofKA

For the case where the concentration of substrate A is held constant,

Eq (7.24) can be expressed as

From determinations of K
at different fixed concentrations of substrate

A, it is possible to obtain estimates ofKA

s andKAB.K
displays a bolic dependence on substrate A concentration (Fig 7.2c) The slope of

hyper-this function equals KA

s KAB In the limit where [A] approaches infinity,

K
= KAB(Fig 7.2c) Thus, by fitting Eq (7.30) to K
–[A] experimentaldata using nonlinear regression, it is possible to obtain estimates of K sA

and KAB

Figure 7.2 Fixed substrate concentration dependence for enzymes displaying ordered

sequential mechanisms: (a) Dependence of Vmax
on [B]; (b) dependence of K
on [B]; (c) dependence of K
on [A].

The dependence ofVmax
on the fixed substrate’s concentration can be used

as an indicator of substrate-binding order A fixed substrate’s tion dependence of Vmax
is associated with the second substrate to bind

concentra-to the enzyme A fixed substrate’s concentration independence ofVmax
isassociated with the first substrate to bind to the enzyme

98 TWO-SUBSTRATE REACTIONS

For this mechanism, the enzyme must bind substrate A first, followed bythe release of product P and the formation of the enzyme species E
This

is followed by binding of substrate B to E
and the breakdown of the

E
B complex to free enzyme E and the second product Q Thus, for pingpong mechanisms, no ternary complex is formed A general steady-statescheme for this type of reactions is

Normalization of the rate equation by total enzyme concentration

(v/[E T]), substitution, and rearrangement yields the following rate

equa-tion for ping-pong bi bi mechanisms:

PING-PONG Bi Bi MECHANISM 99

For the case where the concentration of substrate B is held constant,

Eq (7.37) can be expressed as

displays a hyperbolic dependence on substrate B concentration (Fig 7.3a).

Thus, by fitting Eq (7.39) toVmax
–[B] experimental data using nonlinearregression, it is possible to obtain estimates of Vmax and K mB K
alsodisplays a hyperbolic dependence on substrate B concentration (Fig 7.3b).

Thus, by fitting Eq (7.40) to K
–[B] experimental data using nonlinearregression, it is possible to obtain estimates ofαK mA and K mB

For the case where the concentration of substrate A is held constant,

Eq (7.37) can be expressed as

Figure 7.3 Fixed substrate concentration dependence for enzymes displaying ping-pong

mechanisms: (a) Dependence of Vmax
on [B]; (b) dependence of K
on [B]; (c)

depen-dence ofVmax
on [A]; (d) dependence of K
on [A].

Thus, by fitting Eq (7.42) toVmax
–[A] experimental data using nonlinearregression, it is possible to obtain estimates ofVmaxandαKA

m.K
also plays a hyperbolic dependence on substrate A concentration (Fig 7.3d).

dis-Thus, by fitting Eq (7.43) to K
–[A] experimental data using nonlinearregression, it is possible to obtain estimates ofαK mA and K mB

Differentiation between reaction mechanisms can be achieved by ful scrutiny of the K
versus substrate concentration patterns (Fig 7.4).The adage that a picture tells a thousand words is quite applicable inthis instance It is difficult to determine the mechanism of an enzyme-catalyzed reaction from steady-state kinetic analysis The determination

care-of the mechanism care-of an enzymatic reaction is neither a trivial task nor aneasy task The use of dead-end inhibitors and alternative substrates, study

of the patterns of product inhibition, and isotope-exchange experiments

DIFFERENTIATION BETWEEN MECHANISMS 101

Ping-Pong Random

behavior of one enzyme with n active sites Thus, the rate equation for

an oligomeric enzyme withn independent, noninteracting active sites is

sub-102

SEQUENTIAL INTERACTION MODEL 103

[S]

Figure 8.1 Initial velocity versus substrate concentration curve for a cooperative enzyme.

Enzyme activity can also be affected by binding of substrate and substrate ligands, which can act as activators or inhibitors, at a site other

non-than the active site These enzymes are called allosteric These responses can be homotropic or heterotropic Homotropic responses refer to the

allosteric modulation of enzyme activity strictly by substrate molecules;

heterotropic responses refer to the allosteric modulation of enzyme activity

by nonsubstrate molecules or combinations of substrate and nonsubstratemolecules The allosteric modulation can be positive (activation) or neg-ative (inhibition) Many allosteric enzymes also display cooperativity,making a clear differentiation between allosterism and cooperativity some-what difficult

Cooperative substrate binding results in sigmoidal v versus [S] curves

(Fig 8.1) The Michaelis –Menten model is therefore not applicable tocooperative enzymes Two major equilibrium models have evolved todescribe the catalytic behavior of cooperative enzymes: the sequentialinteraction and concerted transition models The reader should be awarethat other models have also been developed, such as equilibrium associ-ation–dissociation models, as well as several kinetic models These arenot discussed in this chapter

The basic premise of the sequential interaction (SI) model is that cant changes in enzyme conformation take place upon substrate binding,which result in altered substrate binding affinities in the remaining activesites (Fig 8.2) For the case of positive cooperativity, each substratemolecule that binds makes it easier for the next substrate molecule to bind.The resultingv versus [S] curve therefore displays a marked slope increase

signifi-as a function of incresignifi-asing substrate concentration Upon saturation of the

104 MULTISITE AND COOPERATIVE ENZYMES

S S

S

S

Figure 8.2 Diagrammatic representation of the sequential interaction of substrate with

a four-site cooperative enzyme Binding of one substrate molecule alters the substrate affinity of other sites The constantsk depict microscopic dissociation constants for the

first, second, third, and fourth sites, respectively.

active sites, the slope of the curve steadily decreases This results in

a sigmoidal v versus [S] curve (Fig 8.1) For a hypothetical tetrameric

cooperative enzyme with four active sites, the rate equation for the mation of product and enzyme mass balance are

for-v = kcat[ES1]+ 2kcat[ES2]+ 3kcat[ES3]+ 4kcat[ES4] (8.2)

[ET]= [E] + [ES1]+ [ES2]+ [ES3]+ [ES4] (8.3)

The equilibrium dissociation constants, both macroscopic or global (K n)and microscopic or intrinsic (k n), for the various ESn complexes are

Upon substrate binding, dissociation constants can decrease for the case

of positive cooperativity (increased affinity of enzyme for substrate)

or decrease in the case of negative cooperativity (decreased affinity ofenzyme for substrate)

Normalization of the rate equation by total enzyme concentration(v/[E T]), substitution of the different ESn terms with the appropriateexpression containing microscopic dissociation constants, and rearrange-ment results in the following expression for the velocity of a four-site